I'm not sure why you want to prove this by contradiction - your proof is exactly the same as the direct proof, so it adds nothing in terms of understanding. In general, it is inadvisable to use proof by contradiction if a direct proof is just as easily obtained. This is because if you make any mistakes in your working, passing that mistake off as a contradiction gives an invalid proof, even though the conclusion is correct.

For the uniqueness of additive inverse proof, there is a lot of things that may not be well defined. What is (-v)? Is it a third additive inverse? Is it w? Is it u? If (-v) is just an arbitrary additive inverse, does the symbol (-v) represent the same element on both sides? Without knowing uniqueness (which is what you want to prove), how can you add (-v) on each side of the equality without showing they are the same?

There is nothing wrong with proving the method of contradiction, this is essentially the method you use when showing uniqueness. You basically assume two objects possess a certain characteristic and then show that these two objects must in fact be the same. Same reasoning as assuming they are different and finding a contradiction.

However, in your proof (1) you assumed u and w to be the zero vectors and used that assumption to conclude they were the same zero vector, seems a little too easy and too good to be true.

(2) is a good proof except use v^(-1) instead of -v unless you can prove that v^(-1) = -v

There is nothing wrong with proving the method of contradiction, this is essentially the method you use when showing uniqueness. You basically assume two objects possess a certain characteristic and then show that these two objects must in fact be the same. Same reasoning as assuming they are different and finding a contradiction.

However, in your proof (1) you assumed u and w to be the zero vectors and used that assumption to conclude they were the same zero vector, seems a little too easy and too good to be true.

(2) is a good proof except use v^(-1) instead of -v unless you can prove that v^(-1) = -v